As mentioned in the previous post, I’ve been working on designing a microscope to be built on an optical rail. As part of the design, I’ve needed to calculate a bunch of distances and sizes – for instance, the size of the back focal plane – that are not usually provided by the objective manufacturer, but that are easy to calculate. So that you won’t have to hunt down all the necessary formulas (most are in chapter 9 of the Handbook of Biological Confocal Microscopy), I thought I would reproduce them here.

The diagram above shows a schematic of an infinity corrected microscope, consisting of an objective and tube lens. Here are some of the basic parameters of the objective, and how they related to each other and to the ray diagram above:

NA = n sin α

NA: Numerical Aperture of the objective

n: refractive index (1 for an air lens)

α: largest angle the objective can collect

M = f_{t} / f_{o}

M: Objective magnification

f_{o}: Objective focal length

f_{t}: Tube lens focal length (200 mm for Nikon)

FOV = f_{N} / M

FOV: Field of View

f_{N}: Field number (25 mm for Nikon)

d_{BFP} = 2 f_{o} sin α

d_{BFP}: Objective back focal plane diameter

β = arctan (FOV / 2f_{o})

β: Divergence angle of ray bundle

Width of ray bundle at distance d: d_{BFP} + 2d tan(β) = d_{BFP} + d FOV / f_{o}

From this last equation, we can see that putting a stop at some distance from the objective that cuts off part of the ray bundle will result in vignetting of the field of view. This is particularly useful to know when designing your own optical system, so that you don’t accidentally cut off part of the field of view.

With these equations in place, lets calculate them for a Nikon 10x / 0.3 NA objective.

- From the NA, and the fact that this is an air lens (n = 1), we can calculate α = 17.45 degrees.
- From the magnification, and knowing that Nikon uses a 200 mm tube lens, we know that f
_{o}= 20 mm - The field of view is 2.5 mm (from the magnification and the 25 mm Nikon field number)
- The back focal plane diameter is 12 mm
- And β, the beam divergence, is 3.6 degrees.

Useful References:

This post was updated on 4/21/2014 to correct an error in how “d” was drawn in the figure above.

For the objective back focal plane diameter, should it be 2*f*Tan(alpha)? Just asking.

For why the back focal plane diameter is 2*f*sin(alpha) and not tan(alpha), see the last two references in the original post.

You say that it is 25mm for nikon, but i assume it is for their M25 thread objectives, not for their M32 threaded ones. Also, most of the information I have found online about F.N is that it refers to the aperture of the eyepiece. However the following definition “FN is the diameter (in mm) of the image at the intermediate image plane”, tends to indicate that FN is not solely a characteristic of the eyepiece, and either or both the objective and the tube lens can be a limiting factor. So what does field number exactly refer to and can i say it is the minimum of the objective exit pupil and the eyepiece entrance pupil? I am learning a lot from your posts, thank you!

In my comment above I mistakenly omitted to introduce the subject of my question in the first sentence… “You say that the field number is 25mm for nikon” is what I meant to write…

Hi Ludovic,

The field number is a property of the objective lens only. It is the size of the image at the intermediate image plane. So in general your camera sensor should be smaller than the FN.

In recent microscopes with “infinity optical systems” (e.g. Nikon CFI60, Zeiss ICS, etc.) the intermediate image is created by the microscope tube lens. The tube lens could thus limit the FN, but in practice this is never the case.

Hi Kurt,

I just wondered how confident you are of the blanket statement that Nikon objectives have FN = 25 mm. I’ve had trouble getting anything out of Nikon in the past in this regard. Particularly for the high end water dipping objectives (25x/1.1, 16x/0.8) I can imagine that the FN would be a little smaller.

Cheers

I don’t have any special information about the field number of the Nikon objectives – I’ve been going off of what’s posted on MicroscopyU and the Nikon websites. So I would not assign particularly high confidence to that statement.